status of low- fofo snake for final stage of 6d ionization cooling

Neutrino Factory & Muon Collider Collaboration Meeting, Oxford MS January 13-16, 2010

Status of Low- FOFO Snakefor Final Stage of 6D Ionization Cooling

Y. Alexahin (FNAL APC)

Goals 2

Low- HFOFO Status - Y. Alexahin NFMCC Meeting Oxford, MS, January 14 2010

FOFO “snake” to achieve T < 0.5mm, L < 1mm requires:

Small -function at absorbers (at the solenoid center if the phase advance >180/cell

Transverse B-field to generate dispersion, can be created by tilting / displacing the solenoids or by additional dipole coils

Nomenclature:

HFOFO = Helical FOFO channel of alternating solenoids (ASOL)

FOFO-xyz = FOFO with xyz resonance phase advance per focusing cell consisting of one solenoid + all other stuff (RF cavities etc.)

+ – + –

“cell”

FOFO snake for final 6D cooling - original idea

10*By

Bz (T)

x,y [cm]

z

z

Q1, Q2

p/100

orbit lengthening

p/100

- no 2nd order chromaticity,- momentum compaction too small and of the wrong sign!

3


+ –(from Dec.2008 MCDW @ Jlab)

Helical snake for final 6D cooling - the puzzle

By increasing B-field strength it is possible to get phase advance >180/cell and small -function at the solenoid center much smaller emittance.

Tune / period > odd_integer for resonant orbit excitation

Puzzle:

2-cell period (planar snake), Q>1

6-cell period, Q>3

4-cell period, Q>3

6-cell period, Q>5

p < 0

p > 0 significant over-focusing required

4


Naiive 1-dimensional considerations do not work for coupled motion:

besides 180 phase advance of the normal mode there is also 180 rotation of the normal mode

the transverse field phase should change by either 360 or 0 / cell !

Return to FOFO-180°

Actually constant By works just as well!

Since there is no clear gain in going to 270° (just problems), more effort was applied to find solution for 180°. A superposed dipole field generated by additional coils can do the job. Such a field can not be obtained by displacing solenoids:

By / BLSDy

Bz / BLS

x

y

Dx

5


(from Dec.2009 MCDW @ BNL)

FOFO-180° Momentum Acceptance 6

p/p0 – 1

p/p0

p

dK p

])(

)(1[)( 0

Q2 - 1

p/p0

_min [cm]

Q1 - 1

This design is not finished yet

Search for a helical configuration is underway

Emittances well below 0.5mm can be expected


Attempts to equalize cooling rates of the transverse modes were unsuccessful so far

Next Stop - HFOFO-240° 7


4/3 per cell 4 2 per 6-cell period:the resonance By harmonic is 4, but actually the 1rd harmonic is the most efficient!

No big problem with the transverse mode cooling rate equalization:

With solenoid pitch 1.4 mrad, constant quadrupole field of gradient 0.75T/m, 6mm LiH absorbers

mode I II IIItune 0.286+0.0024i 4.384+0.0049i 4.388+0.0048i_eq (mm) 0.74 0.38 0.40

But the momentum acceptance is too small:

Q3

p/p0

Q2

Another difficulty: too high synchrotron tune with 6-cell period: Q1 ~ 0.3

HFOFO-270° 8


2

3

4

1

Original HFOFO-270° channel parameters (4-cell period):

800 MHz pillbox RF 2 8cm, Emax=32MV/m

Solenoids: L=8cm, Rin=16cm, Rout=26cm,Inclination: 3mrad v+ h+ v- h-

Absorbers: LiH, width (on-axis) 1.5cm, no wedge angle

Total length of 4-cell period 1.12m

Problems:

Bzmax=18.5 T (on-axis) for p0=100MeV/c

Limited dynamic momentum acceptance

Large ratio Bz_coil/Bz_axis=2.6 !

3/2 per cell 3 2 per 4-cell period:the fundamental By harmonic is 1, the 3rd harmonic is working

B

r / Rin

2.6 !Bz / Bz0

(V.Balbekov’s formulas used)

Choice of the Solenoid Geometry 9


Bz_coil/ Bz0 (left) and B3/Bz0 (right) vs coil inner radius Rin (cm). - Field on coil drops with smaller Rin, but nonlinearity grows.

Bz_coil/ Bz0 (left) and B3/Bz0 (right) vs coil half-length L/2 (cm). - Both field on coil and nonlinearity decrease with L, but this leaves no room for RF!

Revised HFOFO-270°

With constant quadrupole field of gradient 1.5T/m

mode I II IIItune 0.191+0.0015i 3.254+0.0035i 3.288+0.0036i_eq (mm) 0.62 0.29 0.30

10


600 MHz pillbox RF 2 8cm, Emax=17MV/m (on-axis)

Solenoids: L=16cm, Rin=14.5cm, Rout=38cm (Bz_coil/Bz_axis=1.5)Inclination: 2.24mrad: v+, h+, v-, h-

Absorbers: 2cm LH2 with 1mm LiH windows, no wedge angle

Total length of 4-cell period 4 38cm = 1.52m

Bz_axis=12.6T (Bz_coil=19T) for p0=90MeV/c

z (cm)

100Bx

Bz

100By

B (T)

y

z (cm) x

(cm)

Revised HFOFO-270° Momentum Acceptance

p/p0

Static momentum acceptance is fine: (-9%,+27%),

but the dynamic acceptance is only 25% (full width) – enough to accommodate the equilibrium momentum spread, but not the incoming beam

To increase dynamic acceptance one can:

- reduce solenoid pitch angle (to reduce p)

- reduce p0 (to increase 1/2)

Both these methods are bad for longitudinal cooling –

Careful optimization is necessary

p

dK p

])(

)(1[)( 0

11


Q3

Q2

p0 = 80

p0 = 90

p/p0 - 1

HFOFO-270° G4BL Simulations 12


Precision check: periodic orbitred – MICCD, blue – G4BL

y (cm)

z (cm)

z (cm)

pz (MeV/c)

z (cm)

x (cm)



Phase portrait of initially Gaussian beam: blue – after 1st pass, red – after 16th pass (22.8m)

px, py are mechanical momenta, p0=90MeV/c

py/p0 (p - p0)/p0px/p0

y (cm) t (ns)x (cm)

stochastics on, decays off



Fitted normalized emittances (final values 0.51mm, 0.36mm, 0.35mm),number of muons (N16/N1=908/5222=17.4%) and Palmer’s 6D cooling efficiency1N

2,3N

pass #

pass #

N

Q6

pass #

Total 6D cooling efficiency is just Q6=2: 50% cooling 50% shaving

Summary & Outlook 15


FOFO-180 promises large momentum acceptance, DA and small emittances. However, no way to equalize the transverse mode cooling rates has been found yet. Still, there are some possibilities to explore.

HFOFO-270 can provide emittances T ~ 0.35mm, L ~0.5mm with LH2 absorbers, but the transmission is poor. It can be (probably) improved with nonlinear shape of the absorbers reducing ionization losses for low-momentum muons.

If both efforts fail, retreat below 180: the allowed maximum field of 19T (on the coil) will be achieved with smaller cell length the increase in min and emittances will be moderate

Plans (from MCDW09) 16

Search for the final stage HFOFO configuration (by IPAC10)

Front end with HFOFO (by April 2010 IDS meeting)

Determination of the number of stages and configuration for each stage (by the next MCDW)

Optimization of each stage parameters (2010-2012)

End-to-end simulation with matching between the stages (by 2012)


status of low- fofo snake for final stage of 6d ionization cooling

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